nLab class S field theory




Class 𝒮\mathcal{S} (for “six”) originally denoted a specific class of N=2N=2 4d super Yang-Mills theory introduced in Gaiotto, Moore & Neitzke 2009. They originate from KK-compactification of M5-branes on punctured Riemann surfaces CC and are labelled by a simply laced Lie group?, a Riemann surface CC, and a decoration of the punctures of CC by defect operators.

By further compactifications, one relates theories of this class with certain superconformal field theories in dimension 22 (or, in other formalism, chiral algebras) which are now also said to be of class SS (see at AGT correspondence).


Original articles

see also

In the AGT correspondence:

  • Bruno Le Floch, A slow review of the AGT correspondence, J. Phys. A: Math. Theor. 55 (2022) 353002 doi arXiv:2006.14025

On vertex operator algebras/chiral algebras of class 𝒮\mathcal{S}:

  • Tomoyuki Arakawa, Chiral algebras of class S and Moore-Tachikawa symplectic varieties, arXiv:1811.01577

    We give a functorial construction of the genus zero chiral algebras of class 𝒮\mathcal{S}, that is, the vertex algebras corresponding to the theory of class 𝒮\mathcal{S} associated with genus zero pointed Riemann surfaces via the 4d/2d duality discovered by Beem, Lemos, Liendo, Peelaers, Rastelli and van Rees in physics. We show that there is a unique family of vertex algebras satisfying the required conditions and show that they are all simple and conformal. In fact, our construction works for any complex semisimple group G that is not necessarily simply laced. Furthermore, we show that the associated varieties of these vertex algebras are exactly the genus zero Moore-Tachikawa symplectic varieties that have been recently constructed by Braverman, Finkelberg and Nakajima using the geometry of the affine Grassmannian for the Langlands dual group.

On superconformal theories of class 𝒮\mathcal{S}:

  • Leonardo Rastelli, S. S. Razamat, The superconformal index of theories of class 𝒮\mathcal{S}, in: Teschner, J. (eds) New Dualities of Supersymmetric Gauge Theories. Mathematical Physics Studies. Springer 2015 doi

  • Christopher Beem, Wolfger Peelaers, Leonardo Rastelli, Balt C. van Rees, Chiral algebras of class S, JHEP 05 (2015) 020 doi arXiv:1408.6522

On quantum Seiberg-Witten curves in relation to class S-theories and “M3”-defect branes inside M5-branes:

category: physics

Last revised on January 8, 2024 at 08:37:06. See the history of this page for a list of all contributions to it.